Step |
Hyp |
Ref |
Expression |
1 |
|
sigar |
|- G = ( x e. CC , y e. CC |-> ( Im ` ( ( * ` x ) x. y ) ) ) |
2 |
1
|
sigarval |
|- ( ( A e. CC /\ A e. CC ) -> ( A G A ) = ( Im ` ( ( * ` A ) x. A ) ) ) |
3 |
2
|
anidms |
|- ( A e. CC -> ( A G A ) = ( Im ` ( ( * ` A ) x. A ) ) ) |
4 |
|
cjcl |
|- ( A e. CC -> ( * ` A ) e. CC ) |
5 |
|
id |
|- ( A e. CC -> A e. CC ) |
6 |
4 5
|
mulcomd |
|- ( A e. CC -> ( ( * ` A ) x. A ) = ( A x. ( * ` A ) ) ) |
7 |
|
cjmulrcl |
|- ( A e. CC -> ( A x. ( * ` A ) ) e. RR ) |
8 |
6 7
|
eqeltrd |
|- ( A e. CC -> ( ( * ` A ) x. A ) e. RR ) |
9 |
8
|
reim0d |
|- ( A e. CC -> ( Im ` ( ( * ` A ) x. A ) ) = 0 ) |
10 |
3 9
|
eqtrd |
|- ( A e. CC -> ( A G A ) = 0 ) |